Recently (20th century) the Sets Theory attempted to give a fair basis to the whole logic and mathematics while keeping as free as possible of any reference to peculiar objects, whatever they are. Let us first remind what Sets Theory is.
The Sets Theory is based on three axioms supposed to be obvious universal truths, from which all mathematical truths (calculations, theorems) can be drawn through rigorously exact logical reasoning (demonstration).
1st axiom: Any collection of objects for which there is a criterion making it possible to say if these objects are elements or not of the collection is a set.
2nd axiom: A set cannot be an element of itself.
3rd axiom: The collection of all the sets is not a set.
(Added July 2015, reviewed and moved November 2017: this is the Sets theory I learned in secondary school, and which is discussed in science reviews. I am well aware that wikipedia gives a different description, quite confuse and unconnected to the other parts of mathematics. But I am more confident in the secondary school than in anonymous editors, leaving them the care of explaining themselves publicly on the difference)
(In the Sets Theory as well as in all the following texts, «object» refers to any element on which we shall make logical reasoning, whatever it belongs to concrete domains, abstract domains, ideas, feelings, spirit, or even imaginary)
From these three axioms we can find all the familiar mathematical objects. For example we define the empty set which does not contain any element, and is said of cardinal zero. A set which only contains the empty set is said of cardinal one; a set which contains the empty set more the previous is said of cardinal two, a set which contains the empty set more the two previous is said of cardinal three, and so on we define all the integers. Their relations give the arithmetic operations, addition, subtraction, multiplication, division, then with various rigorous constructions we find the rational numbers, then real numbers... The Aristotelian logic, its AND and its OR, results from the familiar operations of intersection and union of sets. At last the study of the structures of these sets leads to algebra, equations and vector spaces. This notion of an «abstract space» into a set, matching exactly with our everyday life notion of space, will be of a great importance in part three on metaphysics.
I state here that the axioms of the Sets Theory, far from being absolute truths, represent already a given choice, which only selects the results that they are supposed to demonstrate.
The third axiom already arises questioning. No school child ever wondered: why the collection of all the sets would not be a set? The problem is that if we regard it as such, then the second axiom is violated. If we do not regard it as such, then the first is violated. The choice which was however made by mathematicians has good reasons, but there is a feeling as if this third axiom is there only to hide a paradox, a situation where logic is impotent to determine reality. Why simply not to recognize that logic has limits and that it sometimes leads to paradoxes, statements which we cannot show if they are true or false? Instead of stating that this set is not one, it is enough to say that its existence is a paradox. And, after having a glance through the window, we would obtain a rigorous demonstration of a fact: Paradoxes never prevented wheat to grow.
The second axiom also appeared essential to members of the Bourbaki group who formalized the Sets Theory. However some mathematicians remove it, thus leading toward another theory, known as of the Supersets Theory (note 36). At a rough guess, some paradoxes may appear there in quantity, because of circular references, thus bringing back this discussion to that of the third axiom: in the classical Sets Theory, we refuse to consider self-contained sets, to avoid appearance of paradoxes or unsettlable statements. We shall have occasion to speak again of paradoxes throughout this book, and especially in the third part on metaphysics, where we found them a very good practical use.
But the main problem I want to point at now is with the first axiom, to which, as far as I know, no mathematician made any criticism. And yet I see there two completely arbitrary and very interesting a priori:
1) The idea that objects are unavoidably separated and distinct the ones from the others.
2) The idea of absolute criterion for belonging to the set, completely true or completely false, without any nuance.
It is not the matter here to say that the first axiom would be «false», but to make the reader aware that it results from a choice among various possibilities, in favour of special objects which would have the particular properties described above. Only this choice leads to the Sets Theory. Other choices were considered by mathematicians, but they remained poorly studied, as unworkable... for Aristotelian logic!
What I say here is that considering objects with different properties would lead to other theories, and thus to other logics. And I dispute that Aristotelian logic (statements with univocal meaning, without any ambiguity completely true or completely false) «is demonstrated» by the Sets Theory. This logic simply arises from the two choices above; it is implied from the beginning by these two choices. In peculiar the second choice forcefully leads to the idea of completely true or completely false statement, which is the very basis of Aristotelian logic. If we want to demonstrate the theorems of Aristotelian logic and to build mathematics starting from these two choices, the Sets Theory is perfect; but its basic axioms cannot be demonstrated from its results. With my opinion it would be necessary to reformulate the first axiom of the Sets Theory, in order to show that Aristotelian logic directly arises from the two above choices.
Then is Aristotelian logic «true»? Is it «false»? Or does its veracity depends on the direction of the wind or on our goodwill, as claim those who use this kind of arguments biased by self-centredness to dispute ethics and escape its obligations? What I state here is that there is no meaning to say that Aristotelian logic would be absolutely true or absolutely false, but that Aristotelian logic is valid for peculiar objects which satisfy the two above criteria (choice 1 and 2), and only for them. Independently of course of our personal interests, this significant point being discussed in detail in chapter I-8.
Now, what happens in the next cases, with other types of objects? It is what we are goint to see in the next chapter I-3.
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